3.322 \(\int \frac{(a+b x)^{9/2}}{x^6} \, dx\)

Optimal. Leaf size=119 \[ -\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 \sqrt{a}}-\frac{63 b^4 \sqrt{a+b x}}{128 x}-\frac{21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac{21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac{(a+b x)^{9/2}}{5 x^5}-\frac{9 b (a+b x)^{7/2}}{40 x^4} \]

[Out]

(-63*b^4*Sqrt[a + b*x])/(128*x) - (21*b^3*(a + b*x)^(3/2))/(64*x^2) - (21*b^2*(a
+ b*x)^(5/2))/(80*x^3) - (9*b*(a + b*x)^(7/2))/(40*x^4) - (a + b*x)^(9/2)/(5*x^
5) - (63*b^5*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*Sqrt[a])

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Rubi [A]  time = 0.117539, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{63 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{128 \sqrt{a}}-\frac{63 b^4 \sqrt{a+b x}}{128 x}-\frac{21 b^3 (a+b x)^{3/2}}{64 x^2}-\frac{21 b^2 (a+b x)^{5/2}}{80 x^3}-\frac{(a+b x)^{9/2}}{5 x^5}-\frac{9 b (a+b x)^{7/2}}{40 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(9/2)/x^6,x]

[Out]

(-63*b^4*Sqrt[a + b*x])/(128*x) - (21*b^3*(a + b*x)^(3/2))/(64*x^2) - (21*b^2*(a
+ b*x)^(5/2))/(80*x^3) - (9*b*(a + b*x)^(7/2))/(40*x^4) - (a + b*x)^(9/2)/(5*x^
5) - (63*b^5*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*Sqrt[a])

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Rubi in Sympy [A]  time = 15.7766, size = 112, normalized size = 0.94 \[ - \frac{63 b^{4} \sqrt{a + b x}}{128 x} - \frac{21 b^{3} \left (a + b x\right )^{\frac{3}{2}}}{64 x^{2}} - \frac{21 b^{2} \left (a + b x\right )^{\frac{5}{2}}}{80 x^{3}} - \frac{9 b \left (a + b x\right )^{\frac{7}{2}}}{40 x^{4}} - \frac{\left (a + b x\right )^{\frac{9}{2}}}{5 x^{5}} - \frac{63 b^{5} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{128 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(9/2)/x**6,x)

[Out]

-63*b**4*sqrt(a + b*x)/(128*x) - 21*b**3*(a + b*x)**(3/2)/(64*x**2) - 21*b**2*(a
+ b*x)**(5/2)/(80*x**3) - 9*b*(a + b*x)**(7/2)/(40*x**4) - (a + b*x)**(9/2)/(5*
x**5) - 63*b**5*atanh(sqrt(a + b*x)/sqrt(a))/(128*sqrt(a))

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Mathematica [A]  time = 0.0906934, size = 86, normalized size = 0.72 \[ \frac{1}{640} \left (-\frac{\sqrt{a+b x} \left (128 a^4+656 a^3 b x+1368 a^2 b^2 x^2+1490 a b^3 x^3+965 b^4 x^4\right )}{x^5}-\frac{315 b^5 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(9/2)/x^6,x]

[Out]

(-((Sqrt[a + b*x]*(128*a^4 + 656*a^3*b*x + 1368*a^2*b^2*x^2 + 1490*a*b^3*x^3 + 9
65*b^4*x^4))/x^5) - (315*b^5*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a])/640

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Maple [A]  time = 0.018, size = 87, normalized size = 0.7 \[ 2\,{b}^{5} \left ({\frac{1}{{b}^{5}{x}^{5}} \left ( -{\frac{193\, \left ( bx+a \right ) ^{9/2}}{256}}+{\frac{237\,a \left ( bx+a \right ) ^{7/2}}{128}}-{\frac{21\,{a}^{2} \left ( bx+a \right ) ^{5/2}}{10}}+{\frac{147\,{a}^{3} \left ( bx+a \right ) ^{3/2}}{128}}-{\frac{63\,{a}^{4}\sqrt{bx+a}}{256}} \right ) }-{\frac{63}{256\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(9/2)/x^6,x)

[Out]

2*b^5*((-193/256*(b*x+a)^(9/2)+237/128*a*(b*x+a)^(7/2)-21/10*a^2*(b*x+a)^(5/2)+1
47/128*a^3*(b*x+a)^(3/2)-63/256*a^4*(b*x+a)^(1/2))/x^5/b^5-63/256*arctanh((b*x+a
)^(1/2)/a^(1/2))/a^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222017, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, b^{5} x^{5} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (965 \, b^{4} x^{4} + 1490 \, a b^{3} x^{3} + 1368 \, a^{2} b^{2} x^{2} + 656 \, a^{3} b x + 128 \, a^{4}\right )} \sqrt{b x + a} \sqrt{a}}{1280 \, \sqrt{a} x^{5}}, \frac{315 \, b^{5} x^{5} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (965 \, b^{4} x^{4} + 1490 \, a b^{3} x^{3} + 1368 \, a^{2} b^{2} x^{2} + 656 \, a^{3} b x + 128 \, a^{4}\right )} \sqrt{b x + a} \sqrt{-a}}{640 \, \sqrt{-a} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^6,x, algorithm="fricas")

[Out]

[1/1280*(315*b^5*x^5*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) - 2*(965*b
^4*x^4 + 1490*a*b^3*x^3 + 1368*a^2*b^2*x^2 + 656*a^3*b*x + 128*a^4)*sqrt(b*x + a
)*sqrt(a))/(sqrt(a)*x^5), 1/640*(315*b^5*x^5*arctan(a/(sqrt(b*x + a)*sqrt(-a)))
- (965*b^4*x^4 + 1490*a*b^3*x^3 + 1368*a^2*b^2*x^2 + 656*a^3*b*x + 128*a^4)*sqrt
(b*x + a)*sqrt(-a))/(sqrt(-a)*x^5)]

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Sympy [A]  time = 36.351, size = 158, normalized size = 1.33 \[ - \frac{a^{4} \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{5 x^{\frac{9}{2}}} - \frac{41 a^{3} b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{40 x^{\frac{7}{2}}} - \frac{171 a^{2} b^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}}{80 x^{\frac{5}{2}}} - \frac{149 a b^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}}{64 x^{\frac{3}{2}}} - \frac{193 b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{128 \sqrt{x}} - \frac{63 b^{5} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{128 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(9/2)/x**6,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x) + 1)/(5*x**(9/2)) - 41*a**3*b**(3/2)*sqrt(a/(b*x) + 1
)/(40*x**(7/2)) - 171*a**2*b**(5/2)*sqrt(a/(b*x) + 1)/(80*x**(5/2)) - 149*a*b**(
7/2)*sqrt(a/(b*x) + 1)/(64*x**(3/2)) - 193*b**(9/2)*sqrt(a/(b*x) + 1)/(128*sqrt(
x)) - 63*b**5*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(128*sqrt(a))

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GIAC/XCAS [A]  time = 0.215548, size = 147, normalized size = 1.24 \[ \frac{\frac{315 \, b^{6} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{965 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{6} - 2370 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{6} + 2688 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{6} - 1470 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{6} + 315 \, \sqrt{b x + a} a^{4} b^{6}}{b^{5} x^{5}}}{640 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(9/2)/x^6,x, algorithm="giac")

[Out]

1/640*(315*b^6*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) - (965*(b*x + a)^(9/2)*b^
6 - 2370*(b*x + a)^(7/2)*a*b^6 + 2688*(b*x + a)^(5/2)*a^2*b^6 - 1470*(b*x + a)^(
3/2)*a^3*b^6 + 315*sqrt(b*x + a)*a^4*b^6)/(b^5*x^5))/b